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Johannes Anschütz

Publications and source records attributed to Johannes Anschütz.

18 recordsLinked to original sources
arXiv2026

A 6-functor formalism for solid quasi-coherent sheaves on the Fargues-Fontaine curve

We develop a 6-functor formalism $\mathcal{D}_{[0,\infty)}(-)$ with $\mathbb{Z}_p$-linear coefficients on small v-stacks, and discuss consequences for duality and finiteness for pro-étale cohomology of rigid-analytic varieties of general pro-étale $\mathbb{Q}_p$-local systems as well as first examples motivated by a potential $p$-adic analog of Fargues-Scholze's geometrization program of the local Langlands correspondence.

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arXiv2026

Analytic de Rham stacks of Fargues-Fontaine curves

We define and initiate the study of analytic de Rham stacks of relative Fargues-Fontaine curves. To this end, we develop a theory of analytic de Rham stacks with sufficiently strong descent and approximation properties. Specializing to the de Rham stack of the Fargues-Fontaine curve attached to $\mathbb{C}_p$, we apply the general theory to obtain a new geometric proof of the $p$-adic monodromy theorem, avoiding any reliance on earlier results on $p$-adic differential equations. Building on the foundations established here, we plan in a sequel to investigate the cohomology of de Rham stacks of relative Fargues-Fontaine curves in geometric situations and, in particular, provide a stack-theoretic definition of Hyodo-Kato cohomology.

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arXiv2026

On the $p$-adic theory of local models

We prove the Scholze--Weinstein conjecture on the existence and uniqueness of local models for local Shimura varieties, as well as the test function conjecture of Haines--Kottwitz in this framework. To this end, we establish a specialization principle for well-behaved $p$-adic kimberlites, show that these include the v-sheaf local models, determine their special fibers using hyperbolic localization for the étale cohomology of small v-stacks, and analyze the resulting specialization morphism using convolution.

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arXiv2025

Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic

We show that the nearby cycles functor for the $p$-adic Hecke stack at parahoric level is perverse t-exact, by developing a theory of Wakimoto filtrations at Iwahori level, and that it lifts to the $\mathbb{E}_1$-center. We apply these tools to construct the Arkhipov-Bezrukavnikov functor for $p$-adic affine flag varieties at Iwahori level, and prove that it is an equivalence for all classical groups and also exceptional groups of type $E_6$ and $E_7$.

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arXiv2025

v-vector bundles on $p$-adic fields and Sen theory via the Hodge-Tate stack

We describe the category of continuous semilinear representations and their cohomology for the Galois group of a $p$-adic field $K$ with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring $B_{\mathrm{Sen}}$ in a geometric way.

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arXiv2025

Hodge-Tate stacks and non-abelian $p$-adic Hodge theory of v-perfect complexes on rigid spaces

Let $X$ be a quasi-compact quasi-separated $p$-adic formal scheme that is smooth either over a perfectoid $\mathbb{Z}_p$-algebra or over some ring of integers of a $p$-adic field. We construct a fully faithful functor from perfect complexes on the Hodge-Tate stack of $X$ up to isogeny to perfect complexes on the v-site of the generic fibre of $X$. Moreover, we describe perfect complexes on the Hodge-Tate stack in terms of certain derived categories of Higgs, resp. Higgs-Sen modules. This leads to a derived $p$-adic Simpson functor.

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arXiv2023

The small $p$-adic Simpson correspondence in terms of moduli spaces

For any rigid space over a perfectoid extension of $\mathbb Q_p$ that admits a liftable smooth formal model, we construct an isomorphism between the moduli stacks of Hitchin-small Higgs bundles and Hitchin-small v-vector bundles. This constitutes a moduli-theoretic improvement of the small $p$-adic Simpson correspondence of Faltings, Abbes-Gros, Tsuji and Wang. Our construction is based on the Hodge-Tate stack of Bhatt-Lurie. We also prove an analogous correspondence in the arithmetic setting of rigid spaces of good reduction over $p$-adic fields.

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arXiv2022

G-bundles on the absolute Fargues-Fontaine curve

We prove that the category of ``vector bundles on the absolute Fargues--Fontaine curve'' (more precisely the category of sections over some discrete algebraically closed field of the $v$-stack $\mathrm{Bun}_\mathrm{FF}$ of vector bundles on the Fargues--Fontaine curve) is canonically equivalent to the category of isocrystals. We deduce a similar result for ``$G$-bundles on the absolute Fargues--Fontaine curve'' for some reductive group $G$ as well as for sections of $\mathrm{Bun}_\mathrm{FF}$ over classifying stacks for locally profinite groups.

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arXiv2022

Prismatic Dieudonné theory

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DM}^{\rm adm}(R)$ of \textit{admissible prismatic Dieudonné crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DM}^{\rm adm}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

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arXiv2021

A Fourier Transform for Banach-Colmez spaces

We define and study an $\ell$-adic Fourier transform for a relative version of Banach-Colmez spaces (over a perfectoid space which is not necessarily a geometric point), which can be thought of as some analytic analogue of the $\ell$-adic Fourier transform for unipotent perfect group schemes. We explicitly describe it on some examples.

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arXiv2021

Averaging functors in Fargues' program for GL_n

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues' program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm{GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm{GL}_2$ are irreducible. We also attach to any irreducible $\ell$-adic Weil representation of degree $n$ an Hecke eigensheaf on $\mathrm{Bun}_n$, and show, using the local Langlands correspondence and recent results of Hansen and Kaletha-Weinstein, that it satisfies most of the requirements of Fargues' conjecture for $\mathrm{GL}_n$.

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arXiv2021

The $p$-completed cyclotomic trace in degree $2$

We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological cyclic homology $\mathrm{TC}(R;\mathbb{Z}_p)$ of $R$ identifies on $π_2$ with a $q$-deformation of the logarithm.

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arXiv2020

Extending torsors on the punctured Spec(A_inf)

We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine's ring $A_{\mathrm{inf}}$, extend to the whole spectrum. Using descent we can extend a similar result for the ring $\mathfrak{S}$ of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the $B^+_{\mathrm{dR}}$-affine Grassmannian to the Witt vector affine flag variety.

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arXiv2017

Breuil-Kisin-Fargues modules with complex multiplication

We prove that the category of (rigidified) Breuil-Kisin-Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil-Kisin-Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a "p-adic Serre group".

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arXiv2017

Reductive group schemes over the Fargues-Fontaine curve

For an arbitrary non-archimedean local field we classify reductive group schemes over the corresponding Fargues-Fontaine curve by group schemes over the category of isocrystals. We then classify torsors under such reductive group schemes by a generalization of Kottwitz' set B(G). In particular, we extend a theorem of Fargues on torsors under constant reductive groups to the case of equal characteristic.

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arXiv2015

Descent of algebraic cycles

We characterize universally generalizing morphisms which satisfy descent of algebraic cycles integrally as those universally generalizing morphisms which are surjective with generically reduced fibres. In doing so, we introduce a naive pull-back of cycles for arbitrary morphisms between noetherian schemes, which generalizes the classical pull-back for flat morphisms, and then prove basic properties of this naive pull-back.

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